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Function Symmetry Introduction, Transformations of Functions, Algebra 2
This video provides a formal definition of odd and even functions and an intuitive sense of their differences. Sal examines the graphs of both types of functions to show their reflection properties. Functions that are neither odd nor...
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Recognizing Linear Functions, Linear Equations and Functions
Starting with a set of values for the x and y variables, this video shows how to tell whether these points determine a linear or non-linear function. Sal first defines linear functions in terms of the change in y over the change in x,...
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Range, Variance and Standard Deviation as Measures of Dispersion
Continuing his conversation about data sets from the previous video, Sal introduces range, variance, and standard deviation as means of dispersion. Those advancing in algebra or beginning a statistics class will benefit from Sal's clear...
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Pythagorean Theorem II, Right Triangles and Trigonometry, Geometry
This lecture, the second in a series on the Pythagorean Theorem, delves into more complex problems. It primarily examines the process of finding the value of a hypotenuse. The example problem is beneficial because the solution is not...
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Proof: sin(a+b)=(cos a)(sin b)+(sin a)(cos b)
Sal proves the sum of angles formula for sine, sin (a+ b) = cos (a) sin (b) + sin (a) cos (b), through a clear but rather non-intuitive proof. The proof is interesting as it uses the SOH-CAH-TOA model and requires one to recall some...
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Proof: lim (sin x)/x
By looking at the expression (sin x)/x that is not defined at the limit value of 0, Sal uses the squeeze theorem to show that the function can be bound between two other functions that are defined at the limit value, so the limit can...
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Proof: Hyperbola Foci, Conic Sections, Algebra II
In a video that is quite involved, algebraically, Sal proves that the distance of the foci from the center of a hyperbola is the square root of a2+b2. Since the algebra shown here is challenging, this video might be appropriate as an...
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Proof: d/dx(x^n)
Using the binomial theorem and definition of a limit, Sal shows a proof that the derivative of xn equals nxn-1.
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Proof: d/dx(sqrt(x)) Taking Derivatives Differential Calculus
Using the definition of a limit, Sal proves that the derivative of x or x1/2 equals _ x-1/2.
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Proof: d/dx(ln x)=1/x
Using the definition of a limit, various properties of logarithms, and a definition of e, Sal shows the proof of the derivative of ln x = 1/x. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó has a more straightforward...
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Proof: d/dx(e^x) = e6x
Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows proof that the derivative of ex = ex. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó clearly explains this proof.
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Proof: cos(a+b)=(cos a)(cos b)-(sin a)(sin b)
Using a similar method that he used to prove the sum of angles formula for sine, Sal shows the proof of the cosine sum of angles formula, cos (a+ b) = cos (a) cos (b) - sin (a) sin (b). This proof also uses the SOH-CAH-TOA model and some...
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Polar Coordinates 1, Parametric Equations and Polar Coordinates, Precalculus
Sal defines polar coordinates and the relationships between Cartesian coordinates and polar coordinates using basic trigonometric identities and the Pythagorean Theorem in this introductory video.
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Recognizing Number Pattern Examples
This video looks again at finding the algebraic pattern in sequences. In the example, we are looking at a figure where several toothpicks (or lines) are added each time, and you need to find how many toothpicks will be in the 50th...
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Math Patterns Example 1, Applying Mathematical Reasoning, Pre-Algebra
Faced with a series of numbers, we need to find the following few values of the sequence. This short video shows an example of finding the following values in a sequence where we must find the pattern of adding a specific number each...
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Parallel Lines from Equation (Example 2), Mathematics I
Can you tell, algebraically, if three lines are parallel? In this video, Sal demonstrates how to rewrite linear equations in slope-intercept form and compare the slope of each line.
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Introduction to Logarithm Properties (Part 2), Logarithms
Sal gets to the heart of this video, clarifying that it's important to understand math "so you can apply it later on and not have to relearn everything every time." True to his word, Sal demonstrates ways to solve logs and exponents more...
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Introduction to Limits, Limits, Differential Calculus
Sal begins his explanation of limits with a few basic examples and takes a more intuitive point of view before looking at a formal mathematical definition in later videos. He starts by introducing the notation for limits and describes...
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Introduction to Function Inverses, Functions and Their Graphs, Algebra II
Starting from a brief look at functions and the mapping of domains to ranges, Sal starts with an intuitive sense of what a function inverse is. Using an example, he then shows how to find the inverse of a function and how the graph of...
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Intro to 30-60-90 Triangles, Right Triangles and Trigonometry, Geometry
As the fourth part of a series on the Pythagorean Theorem, Sal continues an exploration of the ways to calculate the length of sides of a right triangle using algebra. He also used this theorem to solve problems involving a right...
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Identifying a Hyperbola from an Equation, Conic Sections, Algebra II
In this video, Sal shows again how to write an equation of a conic section in standard form and identify the conic section it represents. This time, his example is of a hyperbola (not centered at the origin), which he also graphs.
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Identifying an Ellipse from Equation, Conic Sections, Algebra II
After watching this video, you should be able to write an equation of a conic section in standard form and identify the conic section from its equation. In an example problem, Sal reviews how to complete the square and graph an ellipse...
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Fun Trig Problem
If solving trigonometric equations is your idea of fun, then this video is correctly titled. Sal uses trigonometric identities, the quadratic formula, and inverse trigonometric functions to solve a trigonometric equation sent in by a...
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Foci of an Ellipse, Conic Sections, Algebra II
The foci of an ellipse are explored in this video. First, foci are shown as the two points on the central axis such that the sum of the distance from a point on the ellipse and the foci points is the same as the distance from any point...